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RAPID COMMUNICATION

Multiple-scattering theory and the relativistic optical model

L. S. Celenza, M. K. Liou, L. C. Liu, and C. M. Shakin

Phys. Rev. C 10, 398(R) (1974) - Published 1 July, 1974

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It is shown how to construct Lippmann-Schwinger equations for the scattering of relativistic projectiles from nuclear targets. The method is based on the Blankenbecler-Sugar reduction of a covariant equation. Among other topics, we discuss the optical model, the relativistic impulse approximation, coupled channel equations, and inelastic scattering.

Expansions for the scattering wave function and radiative capture in $R$ -matrix theory

J. Cugnon and C. Mahaux

Phys. Rev. C 10, 4 (1974) - Published 1 July, 1974

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We derive several expressions for the scattering wave functions in the frame of $R$ -matrix theory. Particular attention is paid to the one-channel situation and to the one-level plus constant background approximation. These results are then applied to the radiative capture process. In particular, we investigate the possible appearance, and the physical interpretation, of asymmetric resonance peaks and of zeros in the capture cross section. The contributions of the external and internal regions are both included. Finally, we show that the resonance parameters of the one-level plus constant background Breit-Wigner approximation are stable against variation of the boundary parameters, the values of which therefore remain largely arbitrary. Correspondingly, the validity of this approximation does not depend critically upon the choice of the boundary parameters.

NUCLEAR REACTIONS $R$ -matrix theory: various expressions for scattering wave functions with application to radiative capture; dependence of resonance parameters upon boundary conditions.

RAPID COMMUNICATION

Electromagnetic properties of low-lying levels of ${}^{129}{Xe}$

G. Marest, R. Haroutunian, I. Berkes, M. Meyer, M. Rots, J. De Raedt, H. Van de Voorde, H. Oonis, and R. Coussement

Phys. Rev. C 10, 402(R) (1974) - Published 1 July, 1974

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Unperturbed and perturbed angular correlation and conversion electron measurements have been performed to obtain spectroscopic information on the low-lying levels of ${}^{129}{Xe}$ fed by the decay of ${}^{129}{Cs}$ . We established $I^{π} (318 keV) = {\frac{3}{2}}^{+}$ , $I^{π} (411 keV) = {\frac{1}{2}}^{+}$ , $I^{π} (589 keV) = {\frac{3}{2}}^{+}$ , $δ (39.6 keV) = - (2.6 \pm 0.5) \times 10^{- 2}$ , $δ (177 keV) = 0.44 \pm 013$ , $δ (372 keV) = + 0.97 \pm 0.09$ , $μ (39.6 keV) = + (0.88 \pm 0.26) μ_{N}$ . $B (M 1)$ and $B (E 2)$ values are derived from above data and earlier lifetime determinations. A weak-coupling calculation is presented to explain experimental data.

RADIOACTIVITY ${}^{129}{Cs}$ [from $I (α, 2 n)$ ]; measured $ce \frac{K}{L}$ , $γ γ (θ)$ , deduced $I$ , $δ$ , $B (E 2)$ , $B (M 1)$ . Measured $γ γ (θ, H)$ , ${}^{129}{Xe}$ level deduced $g$ . Ge(Li), Si(Li) detectors. Isotopically separated source.

RAPID COMMUNICATION

Study of direct-reaction mechanisms and the distorted-wave Born approximation in the $α {(p, d)}^{3}$ He reaction

F. S. Chwieroth, Y. C. Tang, and D. R. Thompson

Phys. Rev. C 10, 406(R) (1974) - Published 1 July, 1974

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A detailed analysis of our calculation on the $α {(p, d)}^{3}$ He reaction using the resonating-group method shows that at sufficiently high energies, the differential reaction cross section can be well accounted for in the forward angular region by a one-nucleon pickup process and in the backward angular region by a two-nucleon pickup process. Also, it is found that even at relatively low energies, the distorted-wave Born approximation or weak-coupling approximation is reasonably valid for this reaction.

NUCLEAR REACTIONS $α (p, d)$ ; calculated $σ (θ)$ . Resonating-group method; coupled-channel calculation.

RAPID COMMUNICATION

${}^{16}O (t, {}^{3}{He}) {}^{16}N$ reaction and the low-lying $T = 1$ levels of mass 16

E. R. Flynn and J. D. Garrett

Phys. Rev. C 10, 409(R) (1974) - Published 1 July, 1974

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The ${}^{16}O (t, {}^{3}{He})$ reaction at an incident energy of 23.5 MeV has been used to study the low-lying levels of ${}^{16}N$ . A comparison of these results with a previous study of the ${}^{16}O ({}^{3}{He}, t) {}^{16}F$ reaction permits assignment of $J^{π} = 0^{-} (1^{-}), 1^{-} (0^{-}), 2^{-} (3^{-}), and 3^{-} (2^{-})$ for the ground, 0.197-, 0.424-, and 0.720-MeV states of ${}^{16}F$ , respectively.

NUCLEAR REACTIONS ${}^{16}O (t, {}^{3}{He})$ , $E = 23.5$ MeV. ${}^{16}N$ levels, analog identification in ${}^{16}O$ , ${}^{16}F$ , deduced $J^{π}$ in ${}^{16}F$ .

RAPID COMMUNICATION

Relation between the optical model and the valence model of $E 1$ neutron capture at and between resonances

A. M. Lane and S. F. Mughabghab

Phys. Rev. C 10, 412(R) (1974) - Published 1 July, 1974

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We show that there are simple relations between the valence model of neutron capture and the optical model. These are derived without recourse to any particular formal framework. The relations show that the predictions of the valence capture model involve the full optical potential and that it is inadequate to use the real part alone.

RAPID COMMUNICATION

High-spin state at 3918 keV in ${}^{26}{Al}$

H. G. Price, P. A. Butler, A. N. James, P. J. Nolan, and J. F. Sharpey-Schafer

Phys. Rev. C 10, 415(R) (1974) - Published 1 July, 1974

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The level at 3918 keV in ${}^{26}{Al}$ has been populated with the ${}^{23}{Na} {(α, n)}^{26}$ Al reaction and shown to have a spin of $7^{+}$ although a $5^{+}$ possibility cannot be completely excluded. The level decays 100% to the $5^{+}$ ground state and has an $E 2$ strength of 6.7 ± 2.2 Weisskopf units if the $7^{+}$ assignment is assumed. Recent calculations with the Glasgow shell model program using the interaction of Preedom and Wildenthal predict a $7^{+}$ state at this excitation energy.

NUCLEAR REACTIONS ${}^{23}{Na} (α, n γ)$ , $E = 8.8$ MeV; measured $σ (θ_{γ})$ , $P$ . ${}^{26}{Al}$ level deduced $J$ , $π$ , $δ$ .

RAPID COMMUNICATION

New projection method and generalized rotational spectrum

Chindhu S. Warke

Phys. Rev. C 10, 418(R) (1974) - Published 1 July, 1974

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Lanczos's algorithm for the matrix eigenproblem is used to project out angular momentum $(J)$ components from a given deformed intrinsic wave function, containing $N$ different values of $J$ . Using this method, it is proved that the low-lying energy spectrum of a nucleus obtained by using the wave functions projected from a deformed intrinsic state (Hartree-Fock, Hartree-Fock-Bogoliubov, or random-phase approximation) has a general form suggested by Bohr and Mottleson in their macroscopic approach, $E (J) = E (β) + Σ \overset{N - 1}{1} A_{n} (β) J^{n} {(J + 1)}^{n} .$ Expressions for $E (β)$ and $A_{n} (β)$ are derived. The first iteration gives Skyrme's formula for the nuclear moment of inertia. The above analytic form of $E (J)$ is exploited to deduce certain conclusions about the nature of the projected energy as a function of $J$ .

RAPID COMMUNICATION

Intermediate resonances in the range of excitation energy from 13.706 to 15.756 MeV in ${}^{30}P$ through the ${}^{28}{Si} (d, p) {}^{29}{Si}$ reaction

C. C. Hsu, T. P. Pai, T. Tohei, and S. Morita

Phys. Rev. C 10, 422(R) (1974) - Published 1 July, 1974

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10 intermediate resonances with their spins and parities were identified in this study. The average width is 160 ± 60 keV, which is consistent with the value 153 keV predicted by Hsu's relationship.

NUCLEAR REACTION ${}^{28}{Si} (d, p)$ , $E = 2.0 - 4.2$ MeV; measured $δ (E; θ)$ ; $θ = 15 - 165 °$ , $Δ θ = 15 °$ ; channel correlation analysis; discussed intermediate structure.

RAPID COMMUNICATION

Calculation of ${}^{3}H$ and ${}^{3}{He}$ electromagnetic form factors with hard - core two - nucleon potentials using the unitary pole approximation

C. Maheshwari and V. S. Mathur

Phys. Rev. C 10, 425(R) (1974) - Published 1 July, 1974

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The triton wave function for a two body spin-dependent hard-core potential of exponential shape is obtained by solving the Faddeev three-body equations in the unitary pole approximation. It is then used to calculate the ${}^{3}H$ , ${}^{3}{He}$ electromagnetic form factors. As anticipated by Gupta et al., the hard core is found to partially account for the difference between the rms charge radii of ${}^{3}{He}$ and ${}^{3}H$ which remains unexplained even after considering the $S^{'}$ state admixture as well as the neutron charge form factor.

NUCLEAR STRUCTURE ${}^{3}H$ , ${}^{3}{He}$ calculated electromagnetic form factors, hard-core exponential potential, UPA calculations.

Erratum: ${}^{207}{Pb} (p, d) {}^{206}{Pb}$ reaction and some matrix elements of the effective interaction

W. A. Lanford and G. M. Crawley

Phys. Rev. C 10, 427 (1974) - Published 1 July, 1974

Interference between Coulomb and nuclear excitation in the inelastic scattering of ${}^{22}{Ne}$ from ${}^{88}{Sr}$

E. E. Gross, H. G. Bingham, M. L. Halbert, D. C. Hensley, and M. J. Saltmarsh

Phys. Rev. C 10, 45 (1974) - Published 1 July, 1974

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We have measured the excitation function at $θ_{lab} = 175 °$ for the excitation of the first $2^{+}$ state in ${}^{88}{Sr}^{*}$ by inelastic scattering of ${}^{22}{Ne}$ . At the same time we obtained similar data for the excitation of the first $2^{+}$ state of the beam particle itself. The measurements span the energy range 49-65 MeV containing the region of the Coulomb-nuclear interference. In terms of Rutherford trajectories, the ${}^{22}{Ne}^{*}$ interference minimum is located at a separation ∼ 0.2 fm greater than for ${}^{88}{Sr}^{*}$ . An optical potential was obtained by a fit to elastic scattering data consisting of an excitation function at $θ_{lab} = 175 °$ and an angular distribution at 65.4 MeV lab. A collective model distorted-wave Born-approximation calculation with this potential can account for the ${}^{88}{Sr}^{*}$ data but fails to describe the shape of the ${}^{22}{Ne}^{*}$ excitation function.

NUCLEAR REACTIONS ${}^{88}{Sr}$ ( ${}^{22}{Ne}$ , ${}^{22}{Ne}$ ), $E = 49 - 65$ MeV; measured $d σ (E)$ at $θ_{1} = 175 °$ for g.s., first $2^{+}$ state in ${}^{88}{Sr}$ and ${}^{22}{Ne}$ , $d σ (θ)$ at 65.4 MeV for g.s.; deduced optical model potential, deformation parameters. Enriched target.

${}^{9}{Be} (p, 2 n) {}^{8}B$ reaction as a neutrino source

N. E. Davison, M. J. Canty, D. A. Dohan, and A. McDonald

Phys. Rev. C 10, 50 (1974) - Published 1 July, 1974

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The ${}^{8}B$ yield from the ${}^{9}{Be} (p, 2 n) {}^{8}B$ reaction has been measured in the range of incident proton energies $E_{p} = 22.0$ to $E_{p} = 47.5$ MeV. The current of 47.5 MeV protons necessary to produce a flux of 0.74 × $10^{6}$ ${}^{8}B$ neutrinos per ${cm}^{2}$ sec (1 solar neutrino unit) at a distance of 8 m from a thick target is found to be 43 ± 5 mA.

NUCLEAR REACTIONS, RADIOACTIVITY ${}^{9}{Be} (p, 2 n) {}^{8}B$ , ${}^{12}C (p, α n) {}^{8}B$ , $E = 22.0 - 47.5$ MeV; measured $σ {}^{9}{Be} (p, 2 n)$ , upper limit $σ {}^{12}C (p, α n)$ ; deduced proton current produce 1 SNU ${}^{8}B$ neutrinos.

Various cross sections for $A <~ 3$ nuclei

Nelson Jarmie and J. H. Jett

Phys. Rev. C 10, 54 (1974) - Published 1 July, 1974

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Differential cross section data, accurate to an error of less than 1%, are presented for several different elastic scatterings and reactions, in the region of 10-14 MeV. The data are for various processes stemming from the bombardment of hydrogen, deuterium, ${}^{3}{He}$ , and tritium by protons and deuterons.

NUCLEAR REACTIONS $A (b, c) D$ , $A = {}^{1}H, {}^{2}H, {}^{3}H, {}^{3}{He}$ , $b = p, d, t$ , $E = 10 - 13.6$ MeV, measured $σ (θ)$ , various angles, $Δ θ = 0.03 °$ , $Δ σ (θ) = 0.5 - 1.0$ .

Polarized protons on ${}^{3}{He}$ at 13.6 MeV

Nelson Jarmie and J. H. Jett

Phys. Rev. C 10, 57 (1974) - Published 1 July, 1974

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Accurate analyzing powers for the scattering of polarized protons by ${}^{3}{He}$ have been measured at 13.600 MeV. They confirm previous measurements and leave a significant difference between $n + {}^{3}H$ and $p + {}^{3}{He}$ analyzing powers at 22 MeV unexplained. The data are currently being used in a general energy-dependent analysis of the $p + {}^{3}{He}$ system.

NUCLEAR REACTIONS ${}^{3}{He} (\vec{p}, p)$ , $E = 13.600$ MeV; measured analyzing power $A (θ)$ , $θ = 15 - 165 °$ , $Δ θ = 0.03 °$ , $Δ A = 0.004 - 0.02$ .

Variational many-channel nuclear reaction formalism

W. Tobocman

Phys. Rev. C 10, 60 (1974) - Published 1 July, 1974

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We present multichannel nuclear reaction formalisms based on the Schwinger and Kohn variational principles. Our formalisms seem somewhat simpler in form than those heretofore proposed. The necessity of using the entire set of asymptotic boundary conditions to constrain the trial wave function is stressed. We show how this may be done by employing the coupled integral equations formalism discussed by Baer and Kouri, Kouri and Levin, and by Tobocman. Several explicit expressions for the transition amplitude are presented which appear to provide a promising basis for doing many-body system scattering reaction calculations.

NUCLEAR REACTIONS Variational multipartition nuclear reaction formalism; coupled LS equation method used to impose all asymptotic boundary condition constraints.

Connected kernel methods in nuclear reactions. II. Rearrangement reactions and the distorted-wave series

Edward F. Redish

Phys. Rev. C 10, 67 (1974) - Published 1 July, 1974

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The newly developed connected kernel equations for $N$ -body scattering are applied to the study of rearrangement reactions. An integral equation for the transition operator is obtained which has a connected kernel and whose inhomogeneous terms contain the standard distorted-wave Born approximation (DWBA) and a two-step part. The two-step part is similar to the expression usually used for the calculation of two-step reactions but with a non-standard prescription for the driving potentials. The difference comes about because of the off-shell transformations used to simplify the original connected kernel equations. A multi-step DWBA series is generated and the conditions for its convergence are discussed.

Neutrons from radiative pion capture in carbon, oxygen, and calcium

W-C. Lam, K. Gotow, B. MacDonald, and W. P. Trower

Phys. Rev. C 10, 72 (1974) - Published 1 July, 1974

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The energy spectra of neutrons between 2 and 28 MeV emitted following radiative pion capture in carbon, oxygen, and calcium fall off exponentially. An enhancement, however, is observed for neutron energies below 5 MeV in carbon and oxygen. Neutron yield peaks at $n - γ$ angles of 180°, indicating a quasifree absorption mechanism in this process.

NUCLEAR REACTIONS ${}^{12}C (π^{-}, n γ)$ , ${}^{16}O (π^{-}, n γ)$ , ${}^{40}{Ca} (π^{-}, n γ)$ , $E = 0$ MeV; measured reaction rate, neutron energy spectrum, and $n - γ$ angular correlations; deduced giant resonance states and direct emission contributions.

Structure of low-lying negative parity states of ${}^{11}B$ and ${}^{11}C$

M. R. Meder and J. E. Purcell

Phys. Rev. C 10, 84 (1974) - Published 1 July, 1974

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The low-lying negative parity states of ${}^{11}B$ and ${}^{11}C$ are interpreted as arising from a $p_{\frac{3}{2}}$ hole coupled to the ground state and first excited state of ${}^{12}C$ . The coupling interaction has been expanded in a series of spherical tensor operators including dipole and quadrupole terms. The parameters of the interaction are found by fitting to the experimental energies, moments, and transition rates of ${}^{11}B$ . The calculated $β$ decay $log ft$ value and most of the calculated static moments and transition rates are in excellent agreement with experiment. The exceptions are attributed to the limited basis used and to the fact that the free core and single particle operators are used; that is no effective operators are used. An improvement in energies and in one transition rate is obtained by treating ${}^{12}C$ as a rotator and including a third core state. Possible improvements in the model are discussed.

NUCLEAR STRUCTURE ${}^{11}B$ , ${}^{11}C$ . Calculated energies, moments, transition rates, $log ft$ , negative parity states.

Multiplet structure of highly excited states in ${}^{15}N$ and ${}^{15}O$

J. E. Purcell and M. R. Meder

Phys. Rev. C 10, 89 (1974) - Published 1 July, 1974

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A series of calculations were performed which support the $α$ -particle core excitation model as a possible explanation of the structure of certain highly excited states seen in ${}^{15}N$ and ${}^{15}O$ . The features of interest are the apparent compression of the core states and the multiplet structure. The compression is semiquantitatively accounted for by an enlargement of the single particle well due to the presence of the $α$ particle. The standard core excitation model fails to account for the multiplet structure. An operator resulting from the nucleon-nucleon spin-orbit interaction is found to account successfully for the multiplet structure. In these calculations, the model of the core is that of a $p_{\frac{3}{2}}$ hole coupled to the states of ${}^{12}C$ . Two different models of the ${}^{12}C$ states are used and both give good results.

NUCLEAR STRUCTURE ${}^{15}O$ , ${}^{15}N$ . Calculated levels, $J^{π}$ . $α$ -nucleon spin-orbit interaction.

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Multiple-scattering theory and the relativistic optical model

Expansions for the scattering wave function and radiative capture in $R$ -matrix theory

Electromagnetic properties of low-lying levels of ${}^{129}{Xe}$

Study of direct-reaction mechanisms and the distorted-wave Born approximation in the $α {(p, d)}^{3}$ He reaction

${}^{16}O (t, {}^{3}{He}) {}^{16}N$ reaction and the low-lying $T = 1$ levels of mass 16

Relation between the optical model and the valence model of $E 1$ neutron capture at and between resonances

High-spin state at 3918 keV in ${}^{26}{Al}$

New projection method and generalized rotational spectrum

Intermediate resonances in the range of excitation energy from 13.706 to 15.756 MeV in ${}^{30}P$ through the ${}^{28}{Si} (d, p) {}^{29}{Si}$ reaction

Calculation of ${}^{3}H$ and ${}^{3}{He}$ electromagnetic form factors with hard - core two - nucleon potentials using the unitary pole approximation

Erratum: ${}^{207}{Pb} (p, d) {}^{206}{Pb}$ reaction and some matrix elements of the effective interaction

Interference between Coulomb and nuclear excitation in the inelastic scattering of ${}^{22}{Ne}$ from ${}^{88}{Sr}$

${}^{9}{Be} (p, 2 n) {}^{8}B$ reaction as a neutrino source

Various cross sections for $A <~ 3$ nuclei

Polarized protons on ${}^{3}{He}$ at 13.6 MeV

Variational many-channel nuclear reaction formalism

Connected kernel methods in nuclear reactions. II. Rearrangement reactions and the distorted-wave series

Neutrons from radiative pion capture in carbon, oxygen, and calcium

Structure of low-lying negative parity states of ${}^{11}B$ and ${}^{11}C$

Multiplet structure of highly excited states in ${}^{15}N$ and ${}^{15}O$

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